1. Introduction

Scanning near-field optical microscopy is one of the best methods to achieve optical superresolution without using a fluorescent markers, [1–4] where the tip design is crucial in achieving better optical and topographical resolutions and improving image contrast. Many different tip designs have been proposed over the years, for example, antenna-on-probes; [5–11] particle-on-probes such as fluorescent beads, [12] nitrogen vacancies, [13,14] or quantum dots; [9,15] plasmonic tips; [16] or plasmonic superfocusing or nanofocusing tips. [3,4,17–20] A real contender among these is the superfocusing tip, favored by many groups and successfully demonstrated in Raman [3,21,22] spectroscopies and other near-field measurements. [4,19,23] Many analytical and numerical studies have demonstrated that with the superfocusing effect [24–26] one achieves 3 dimensional electromagnetic field confinement with several orders of magnitude field enhancement at the sharp metallic tip apex of metallic wedges or cones. [27,28] However, there are only few experimental proofs of the phenomenon. The phonon polariton nanofocusing has been measured along the propagation direction in the mid-infrared spectral region (∼ 6.7 µm) in a tapered hexagonal boron nitride layer. [29] There have been several attempts in experimentally realizing the superfocusing in the visible spectrum in various tapered structures, [30–37] and only the light emission from the tip apex was measured and analyzed as an indirect indication of the superfocusing effect. Nevertheless, in the visible spectral range, the electromagnetic field confinement during propagation, namely the gradual and adiabatic decrease of the effective wavelength and mode volume of surface plasmon polaritons, has not yet been measured and experimentally demonstrated in adiabatically decreasing metallic structures, such as metallic cones or wedges. In this letter, we aim to experimentally measure the 2-dimensional electromagnetic field confinement in gold wedges in the visible spectrum. Many numerical studies have proved and illustrated that the superfocusing effect takes place in metallic wedges. Here, we would like to take a look at this problem from a modal perspective as it provides intuitive understanding of the underlying physics. Especially, we want to understand which mode is the superfocusing mode, why this particular mode enables the superfocusing effect, how this superfocusing effect depends on the frequency, what is the optimal operating laser wavelength for the largest field confinement, and what are the experimental conditions for which this effect can be directly observed.

2. Theoretical background

We start our study by exploring a gold wedge structure on a glass substrate and investigate analytically and experimentally the modes and their behavior during propagation in a gold wedge. The gold wedge is illustrated as an inset in Fig. 1(a) where the gold wedge is invariant along y-axis and its height decreases along z-axis. The SPP is excited at the tall edge of the gold wedge on the right hand side and propagates in the direction toward the tip along x-axis. Since the metal layer thickness is slowly varying due to a small wedge angle $\alpha $ ($< 10^\circ $ so that $\sin \alpha \approx \alpha $), we can assume that the SPPs propagate adiabatically through the wedge toward the tip. This assumption allows us to consider an insulator-metal-insulator (IMI) structure to study adiabatically propagating SPPs and their behavior in a metal wedge depending on the wedge height d. By using Maxwell's equations and the corresponding interface conditions in an IMI structure, the dispersion relation is found as [38]

 

Fig. 1. Surface plasmon polaritons (SPP) on a gold wedge. (a) The real part of the propagation constant of the short range (SR, in red) and the long range (LR, in blue) SPP mode depending on the wedge height d. The gold wedge is illustrated as an inset. (b) The imaginary part of the propagation of the short range (SR, in red) and the long range (LR, in blue) SPP mode depending on the wedge height d. The calculation parameters are: vacuum wavelength λ₀ = 633 nm, gold permittivity ε_gold = ‒10.59 + 1.27i, air permittivity ε_air = 1, and substrate (SiO₂) permittivity ε_subs = 2.12.

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Fig. 2. The real part of longitudinal field profiles of SR-SPP (red) and LR-SPP (blue) modes plotted along the transverse z-axis at a fixed wedge height. Solid and dashed lines are for the wedge height d of 100 nm and 10 nm, respectively.

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The LR-SPP mode is extensively studied and widely used in various sensor and modulator applications. [40–43] As Fig. 1(a) shows, the real part of the normalized propagation constant is about 1.05 for the LR-SPP mode at the wedge height of 100 nm, and it changes only slightly for wedge heights ranging between 50 ‒ 100 nm. Furthermore, the imaginary part of the propagation constant as presented Fig. 1(b) is also small compared to the SR-SPP mode (3.5 times at a wedge height of 100 nm) so the LR-SPP mode propagates a much longer distance than its counterpart the SR-SPP, until it decays due to absorption by the metal and leakage into the substrate. As it can be seen in Fig. 2, not only the penetration depth of the fields in the air is long but also the amplitude of the LR-SPP mode is larger in the air than in the substrate. Hence, Fig. 1 and Fig. 2 imply that the excitation efficiency of the LR-SPP mode is relatively high from the air side based on two facts. First, the propagation constant is closer to 1 that is the refractive index of the air. Thus for phase matching, it requires only little extra momentum to excite the LR-SPP mode. Second, since a large portion of the mode energy is guided in the air, the field overlap integral is large giving a large coupling efficiency. Another notable characteristics of the mode is that the effective index only increases slightly while the imaginary part decreases and converges to zero with the decreasing height of the wedge. This is because the LR-SPP mode experiences cut-off at a wedge height of about 50 nm for a vacuum wavelength of 633 nm, and the mode slowly leaks into the substrate and radiates to the far-field. Hence, the mode cannot exist and be excited if the wedge height is smaller than 50 nm. This mode cut-off property can be exploited to individually excite the LR-SPP or SR-SPP mode from free space on a gold wedge. Furthermore, when the metal layer thickness becomes large, the SPPs at the air/metal interface and at the substrate/metal interface are no longer coupled. Hence, toward the larger metal layer thickness, the propagation constants of LR- and SR-SPPs converge to that of SPPs at each interface and no longer sensitive to the metal layer thickness.

Meanwhile, the SR-SPP mode is highly lossy. So the propagation length is short as the name implies. Despite the high propagation loss, this mode behaves in a very interesting way. Just like the superfocusing of the radially polarized SPP mode on a conical metallic structure, [20] the plasmon superfocusing occurs for this SR-SPP mode propagating in a 2 dimensional gold wedge structure. As the mode propagates toward the tip of the wedge, the real part of the propagation constant (or effective index) of the mode increases exponentially resulting in a decrease of the effective wavelength. This means that the electromagnetic field of the mode becomes increasingly compressed longitudinally along the propagation direction with decreasing wedge height. This effect is quite pronounced below a wedge height of 20 nm for a vacuum wavelength of 633 nm as it can be seen in Fig. 1(a) (red curve). Furthermore, the field distribution of the mode goes through a substantial change as it propagates toward the tip of the wedge. The field distributions (Re[E_x]) of the SR-SPP (red lines) modes are calculated along the z-axis at wedge heights of 100 nm (solid red line) and 10 nm (dashed red line) which are shown in Fig. 2. If we examine the penetration depth of fields, it decreases in the substrate and increases in the gold for decreasing wedge height d. This observation suggests that the field becomes transversally confined to the metal layer more and more when approaching the tip in addition to the longitudinal confinement that comes from the shrinking effective wavelength. Moreover, the field profile becomes symmetric, despite the structural asymmetry, on the both sides of the metal as it can be observed in Fig. 2 (dashed red line). This symmetric and in-phase electromagnetic field drives the free electron oscillation at both metal surfaces in unison, and this is one of the main reasons for the superfocusing field enhancement of the SR-SPP mode. All in all, the electromagnetic field is confined transversally and longitudinally while the plasmonic field amplitude increases toward the tip of the wedge. This whole phenomenon is called plasmon superfocusing effect, and it has a great potential in enhancing linear and non-linear light matter interaction by proving highly localized and strong electromagnetic fields at the tip. [27,44]

The plasmonic superfocusing effect is strongly dependent on the light frequency (or the corresponding vacuum wavelength) as the metal is highly dispersive. To understand this relation, we take a look at the dispersion relation (Eq. (1)) depending on the wedge height d and the vacuum wavelength ${\lambda _0}$ of the light. If one ultimately wants to efficiently slowdown the SR-SPP mode, the real part of the propagation constant should increase as early as possible (at a relatively large wedge height) at a fast rate with respect to the decreasing wedge height. Hence, at a wedge height below a certain threshold ($d < {d_{\textrm{Th}}}$), we can assume that the propagation constant is so large that $\frac{{\beta ({\omega ,d} )}}{{{k_0}}} \gg \sqrt {|{{\varepsilon_{\textrm{gold}}}} |} $. Since the permittivity of gold is the largest among the constituent materials of the wedge, the transverse k-vectors (${k_\ast }$) in Eq. (1) can be factored out. Under this assumption, we can simplify Eq. (1) into the following linear equation

Here, we define a new parameter $\eta (\omega )={-} 0.5\ln \frac{{({{\varepsilon_{subs}} + {\varepsilon_{gold}}(\omega )} )({{\varepsilon_{surr}} + {\varepsilon_{gold}}(\omega )} )}}{{({{\varepsilon_{subs}} - {\varepsilon_{gold}}(\omega )} )({{\varepsilon_{surr}} - {\varepsilon_{gold(\omega )}}} )}}$ and call it the superfocusing material parameter. [45,46] This parameter is complex valued $\eta (\omega )= \eta {^{\prime}}(\omega )+ i{\ast }\eta {^{\prime\prime}}(\omega )$. The real part of this parameter expresses the strength of the longitudinal confinement depending on the light frequency and material constituents of the wedge structure while the imaginary part $\eta {^{\prime\prime}}(\omega )$ describes the absorption of the SR-SPP mode. Thus, the superfocusing material parameter can provide us with a simple and intuitive understanding of the superfocusing effect and its dependence on the vacuum wavelength and constituents material. Here, the threshold height ${d_{\textrm{Th}}}$ indicates the height below which the propagation constant $\beta $ is inversely proportional to the height and thus defines the required wedge height to achieve a certain degree of the superfocusing effect for the given wavelength. Hence, Formula 2 holds when the wedge height d is smaller than the threshold height ${d_{\textrm{Th}}}$. We calculate the real part of the normalized propagation constant depending on the wedge height d and the vacuum wavelength ${\lambda _0}$ by using Eq. (1) and plot it in Fig. 3(a). The black solid line indicates the threshold wedge height${\; }{d_{\textrm{Th}}}$ for the given vacuum wavelength where the normalized propagation constant becomes large enough so that Formula 2 is valid. In this case, $|{{\raise0.7ex\hbox{${{\varepsilon_{\textrm{gold}}}}$} \!\mathord{\left/ {\vphantom {{{\varepsilon_{\textrm{gold}}}} {{\beta^2}/k_0^2}}} \right.}\!\lower0.7ex\hbox{${{\beta^2}/k_0^2}$}}} |$ is arbitrarily set to be 0.2. In the figure, one can see that the propagation constant increases toward the green spectral region as well as toward the decreasing wedge height. The threshold wedge height that is illustrated by a black solid line in Fig. 3(a) tells us that in the green spectral region, the superfocusing effect starts to take place at a large wedge height d and more efficiently than in the blue and red spectral regions. Figure 3(b) illustrates the superfocusing material parameter (solid line) and the accumulated propagation loss(dashed line) after the propagation through a wedge for a distance of about 115 nm depending on the vacuum wavelength ${\lambda _0}$. The propagation distance corresponds to a distance from a wedge height of 20 nm to a wedge height of 10 nm when the wedge angle is 5°. The real part of the superfocusing material parameter ($\eta {^{\prime}}$) peaks around the wavelength of ${\lambda _0} = {\; }$526 nm and slowly decrease away from this point toward blue and red regions just like the threshold wedge height. Both of these trends tell us that toward the green region where the gold absorption is the highest, the effect of the longitudinal confinement becomes the strongest and the field confinement starts to take place efficiently and earlier at a larger wedge height than in the red or blue spectral regions.

 

Fig. 3. Wavelength dependence of the superfocusing in a gold wedge. (a) Real part of the normalized propagation constant (${\boldsymbol \beta ^{\prime}}/{{\boldsymbol k}_0}$) depending on the wedge height ${\boldsymbol d}$ and the vacuum wavelength ${{\boldsymbol \lambda }_0}$. The black solid line indicates the threshold wedge height ${{\boldsymbol d}_{{\boldsymbol Th}}}$ for the given excitation wavelength. (b) Superfocusing material parameter and the accumulated propagation loss after a propagation through a wedge for a distance of about 114 nm (from a wedge height of 20 nm to 10 nm when the wedge angle is 5°) that are drawn in a solid line and a dashed line, respectively. The arrows indicate the corresponding vertical axes.

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One can intuitively understand this strong and efficient superfocusing effect as an efficient increase of the energy density of the SPP mode that is achieved by having a significant portion of the mode energy in the metal region. [47,48] As we have shown in Fig. 2, the relative energy density in the metal increases and becomes almost symmetric toward the tip because of the greatly increased propagation constant. This in-phase and symmetric electromagnetic field in the metal drives further the free-electron oscillation in the metal resulting in an increase of the relative electric field and also the Poynting vector in the metal. Consequently, the energy density increases in the gold wedge with the decreasing wedge height d. This results in an increase of the SR-SPP mode’s effective index and hence a decrease of the phase velocity. The Poynting vector of the SPP mode in the metal region is proportional to the propagation constant and proportional inversely to the metal permittivity. [38,49] Hence, the superfocusing effect becomes maximum around the vacuum wavelength of 526 nm for a gold wedge on a glass substrate. Consequently, due to the dispersion of the gold, the superfocusing effect has a different strength depending on the vacuum wavelength, and the favorable condition is fulfilled near the green spectral region.

Although the superfocusing effect is shown to be strong in the green spectral region for a gold wedge, the propagation loss also increases in this spectral region due to the increase in the imaginary part of the propagation constant. Therefore, the strength of the superfocusing effect is not the only parameter for optimization. We would like to estimate the propagation loss depending on the vacuum wavelength. Since the wedge structure has a varying height along the propagation, the conventional calculation of a propagation length ($L \approx 1/\textrm{Im}[{\beta (d )} ]$) cannot be used. Hence, we calculate the accumulated propagation loss of the SR-SPP mode after it propagates from a wedge height of 20 nm down to 10 nm for a wedge angle of 5^○ and present the result in Fig. 3(b) in a dashed line. This distance corresponds to a propagation distance of about 115 nm. Within this short distance of propagation, the SR-SPP mode intensity is reduced about 100 times more for the vacuum wavelength of 526 nm compared with the wavelength of 700 nm. This tells us that one always needs to keep in mind that the superfocusing effect consist of two competing effects: first, the longitudinal confinement caused by the effective wavelength decrease determined by $\eta {^{\prime}}$ and second, the attenuation (losses, absorption) determined by $\eta {^{\prime\prime}}$. Based on this result, we choose to work at a vacuum wavelength of 633 nm in our experimental study. Here, the losses are relatively low so that the propagation length is larger than at 526 nm while the longitudinal confinement is stronger than at 700 nm.

3. Experimental results and discussion

We employ a pseudo-heterodyne scattering scanning near-field optical microscope (SNOM) to measure the SPP field amplitude on a gold wedge structure. The experimental setup is illustrated in Fig. 4 and is explained in detail in the Method section. As illustrated in the figure, a laser beam at a wavelength of 633 nm is focused onto the SNOM tip by a parabolic mirror with a numerical aperture of about 0.3, and the focus spot is about 5 µm. The position of the focus spot and the SNOM tip is fixed while the sample stage holding the gold wedge moves and scans during measurements. Since the focus spot is large, the tail of the focused beam still covers the tall edge of the gold wedge, even when the SNOM tip moves away from the edge of the wedge, enabling the SPP excitation at the edge. The SNOM tip scatters the near field which is collected by the same parabolic mirror and is detected by a photodiode. We fabricated first many identical gold wedges with maximum heights of 200 nm for our experiments, and from these wedges, we tailor wedges with different heights by using the focused ion milling. The average wedge angle is about 0.5°, but this angle varies locally along the propagation direction of the SPP. Hence, in different measurements, we refer to different average wedge angle that corresponds to the region where the SPP has propagated and measured. The detail of the fabrication process is explained in the Method section.

 

Fig. 4. Pseudo-heterodyne scanning near-field optical microscopy setup for detecting the SPP field on a gold wedge. Since the illumination spot is large (∼ 5 µm in diameter), the edge of the gold wedge during scan remains partially covered by the illumination spot of the incident beam and thus enables the SPP excitation. The near-field signal is scattered by the SNOM tip, collected with the same parabolic mirror used for excitation, and directed toward the photodiode. The interference between the reference beam and the scattered light from the SNOM tip is detected with a photodiode and a lock-in amplifier.

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In the experiment, we measure the absolute amplitude (${A_{\textrm{meas}}}$) and the phase (${\varphi _{\textrm{meas}}}$) for each harmonics of the modulated signal. Because we are measuring the SPP field that is excited by the incident field hitting the tall edge of the wedge structure, there remains some incident light contribution in the total measured signal. The scattered field (${E_{\textrm{scat}}} = {\alpha _{\textrm{eff}}}({{E_0} + {E_{\textrm{SPP}}}} )$) by the SNOM tip is a local field at the position of the SNOM tip that is modified by the polarizability of the SNOM tip (${\alpha _{\textrm{eff}}}$). Meanwhile, the local field is a combination of the constant incident field (${E_0}$) at the SNOM tip and the SPP field (${E_{\textrm{SPP}}}$) that reaches the SNOM tip after it is excited at the tall edge of the wedge by the incident field. For a given incidence angle ($\theta $) and a SNOM tip position ($x$) with respect to the edge, we can express the scattered field by the SNOM tip as [50]

In this formula, the ${\varphi _{\textrm{meas}}}$ and the incidence angle $\theta $ play a crucial roles while ${f_0}$ only scales the field amplitude. The incidence angle is fixed in the experimental setup alignment but can be fine-tuned. Sometimes, the phase jumps due to environmental changes such as mechanical vibrations, air fluctuations, acoustic noise etc.. In this case, the phase jump should be numerically removed, and the phase amplitude can also be fine-tuned in the above equation. Furthermore, we assume the SPP excitation by the SNOM tip itself is negligible because the interference between the tip- and wedge edge- launched SPPs and the incidence beam should result in smaller modulation patterns which we did not observe in our measurements. [51]

First, to excite the LR-SPP mode, we choose a wedge with an edge height of about 70 nm. At this height, mostly the LR-SPP is excited because the excitation efficiency of this mode is much higher than the SR-SPP mode due to the better field overlap between the SPP mode and the incident field (see Fig. 2). Optical and topographical measurements obtained by SNOM are presented in Fig. 5(a)-(c), where the green curves show the cross-sections along the horizontal axis, and the scale bar in Fig. 5(a) applies to all 3 images. In Fig. 5(a), the gray region is the gold layer, and the gradual darkening of it indicates the decrease of the gold layer thickness as it can be seen from the green cross-section curve while the black region on the right is the glass substrate. It also can be seen in the scale bar in Fig. 5(a) that the substrate height is below 0. The reason is that during the focused ion beam (FIB) milling and structuring the edge of the wedge, the glass substrate was over-milled creating a trench nearby the edge. This has been verified by scanning a large area around the wedge in a separate experiment (not presented here), and the ground level (glass/gold interface) was determined. Furthermore, the fringe spacing of the interference pattern in Fig. 5(b) is large compared with that of the incidence light and the SPP mode. This is due to the decreasing wavevector in the modulation pattern (${k_{\textrm{modulation}}} = {k_{\textrm{SPP},x}} - k\textrm{cos}(\theta ))$ that is apparent in Eq. (3). Nevertheless, it contains all the information on the SPP mode’s wavevector. In the optical near-field measurement, both 3^rd and 4^th harmonic modulated signals from the tip were measured. Detailed information is given in the Method section. Although the 4^th harmonic signal contains less background contribution, it suffers from poor signal to noise ratio (SNR) resulting in a large numerical error in Formula 4. Meanwhile, the 3^rd harmonic optical signal gives good enough background suppression and SNR and thus contains sufficiently rich information of the SPP field. Therefore, we present here the 3^rd harmonic optical signals and use them in our analysis as well. The extracted SPP field real part of the LR-SPP mode is obtained with Formula 4 and presented in Fig. 5(d) while the averaged field profile along the horizontal axis is plotted in Fig. 5(e). In Fig. 5(e), the dotted black and solid blue lines correspond to the measured data and the theoretically calculated SPP field profile (${E_{\textrm{LR} - \textrm{SPP}}}\sim Re\left[ {\textrm{exp}\left( {i\mathop \smallint \limits_0^L {\beta_{\textrm{LR} - \textrm{SPP}}}({h,x} )dx} \right)} \right]$ where L is the propagation distance), respectively. From the measured data, we extract the effective LR-SPP wavelength ${\lambda _{\textrm{eff}}}$ of 607.90 nm while the theoretically estimated value is around 601.74 nm. Due to the photoresist residues that remained from the fabrication process (see Method section), the sample surface is covered by small particles. The slight deviation in the period could be due to these small particles on the surface that disturb the optical measurement by causing speckles in Fig. 5(b). Despite using median filter to remove this non-uniformity, the optical measurement is slightly disturbed. The comparison of experimental and theoretical results, shown in Fig. 5(d), demonstrates that the LR-SPP mode’s behavior can be well predicted, and for thick gold layers, that the Brendel-Bormann model [39] of gold permittivity describes the gold properties and the SPP field quite well. Note that the theoretically estimated SPP field does not include radiate leakage loss which results in difference between the measured and calculated results.

 

Fig. 5. SNOM measurement of the LR-SPP mode. (a-c) Topography, optical amplitude and phase images, respectively. The optical measurements correspond to the 3^rd harmonic order of the pseudo-heterodyne SNOM. (d) Extracted SPP field real part $Re[E_{ext, SPP} ]$. (e) Averaged field profile from (d) where the solid blue line is the theoretically calculated SPP field profile, and the dotted line is the experimentally measured field profile.

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Second, the SR-SPP mode was experimentally studied at a smaller wedge where the edge height is only about 28 nm, and the average wedge angle is about 1° (in the region where the SR-SPP is measured). For this small wedge, we successfully avoid the excitation of the otherwise dominant LR-SPP mode because this mode experiences cut-off at a wedge height of 50 nm. We measured the topography and optical near-field signals (amplitude and phase) and present the results in Fig. 6(a)-(c). Here, the green curves show the profiles along the horizontal axis averaged over the vertical axis, and the scale bar in Fig. 6(a) applies to all 3 images. Like in the previous case, the glass substrate was over-milled on the right side during the FIB fabrication creating a trench dug below the original surface of the substrate as it can be seen in Fig. 6(a). A large area around the wedge was scanned to establish the substrate surface level (glass/gold interface) in a separate experiment (not presented here). Towards the left side, the sharp edge of the wedge, it can also be seen that the (electro-resistive) continuity of the gold layer is broken, and the gold deposition led to the formation of separated islands rather than a closed layer. Moreover, it can be seen in the optical images in Fig. 6(b) and (c) that the fringe spacing is significantly smaller than that in Fig. 6(b) and (c). This is because the SR-SPP mode’s wave vector is significantly larger than that of the LR-SPP therefore increasing the modulation frequency ${k_{\textrm{modulation}}}$ or reducing the fringe spacing. With the measured total amplitude and phase, the SR-SPP mode’s field real part is extracted with Formula 4 and presented in Fig. 6(d) and (e), where the black dotted line in Fig. 6(e) shows the averaged field profile along the horizontal axis of Fig. 6(d). As the measurement in Fig. 6(e) shows, we managed to measure the first 5 half-cycle of the SR-SPP propagation despite the high absorption loss and the short propagation length. The very first half-cycle has a length of $a = 167.6$ nm, and as the SR-SPP mode evolves through the wedge, the subsequent half-cycles reduce in length from a to 0.75a, 0.62a, 0.54a, and finally to 0.53a as it is shown in Fig. 6(e). At the fifth half-cycle, the length of the half-cycle is reduced by 47% compared with the first one. This corresponds to about 3.5 times reduction of the vacuum wavelength (${\lambda _0} = $633 nm) or an effective index ($\beta ^{\prime}/{k_0}$) of about 3.5. This is a clear demonstration of the longitudinal confinement of the electromagnetic field or the superfocusing effect in a gold wedge. It is important to note that this value is, by no means, the limit of the superfocusing effect in a gold wedge. Rather, we were only able to measure the longitudinal field confinement up to this point due to our measurement limitations. The SNR is not good enough to resolve the SPP field beyond this point. The electromagnetic field is expected to propagate further, and the longitudinal confinement should increase as well. Furthermore, one can see that beyond the propagation distance of 600 nm, the extracted SPP field amplitude increases in Fig. 6(e). This is not physical and is rather a numerical error that is created in Formula 4 when the signal amplitude is too low. Moreover, when the SR-SPP field amplitude is theoretically calculated (${E_{\textrm{SR} - \textrm{SPP}}}\sim Re\left[ {\textrm{exp}\left( {i\mathop \smallint \nolimits_0^L {\beta_{\textrm{SR} - \textrm{SPP}}}({h,x} )dx} \right)} \right]$) for the experimentally measured wedge profile, it was shown that there is a mismatch in the measured height. Theoretically, we expect that the wedge height at the beginning is 21 nm instead of 28 nm to achieve the same amount of longitudinal confinement. This overestimation of the wedge height in the measurement most likely arises from the small particles that are likely the reminiscent of photoresist and are scattered over the gold layer and the glass substrate. Since these non-metallic particles have sizes ranging between 2-10 nm, they make the average measured height of the wedge appear to be large and slightly shifts the gold layer level. Furthermore, a slight mismatch between theoretically estimated and measured height could be also coming from the material model that is commonly used to describe the bulk gold. For a thin gold layer, the bulk material model may not suit well as it oversimplifies the nonlocality of the metal’s response, dissipation, dephasing effect near surface, tunneling etc.. [52,53] Finally, there is another aspect that was not included in the theory yet, which could have a non-negligible effect on the estimated value of the effective index of the SR-SPP mode. A thin Ti-adhesive layer (3 nm) was deposited on the glass substrate before the gold deposition. Although such Ti-deposition usually forms isolated islands rather than a continuous film, an effective medium could form and influence the SR-SPP mode since the mode’s effective wavelength is already quite short (about 167 nm or below). The effective medium created by Ti-islands can improve further the longitudinal field confinement of the SR-SPP mode.

 

Fig. 6. SNOM measurement of the SR-SPP mode. (a-c) Topography, optical amplitude and phase images. The optical measurements correspond to the 3^rd harmonic order of the pseudo-heterodyne detection. (d) Extracted SPP field real part $Re[E_{ext, SPP} ]$. (e) Averaged field profile from (d) where the solid blue line is the theoretically calculated SPP field profile, and the dotted line is the experimentally measured field profile.

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Besides measuring the decreasing effective wavelength, it would be very interesting to measure the decay length of the fields depending on the wedge height by doing an approach curve at each scanning point. This would enable us to estimate not only the field enhancement at tip of the wedge but also the mode volume. However, these are out of scope of the current article and of interest to our future work.

4. Methods

4.1 Fabricating gold wedges

To fabricate the gold wedges, a 14 µm thick layer of positive tone photoresist (AZ 9260) was applied to a glass substrate and exposed with a grating pattern by means of photolithography. After development, the pattern was used as a shadow mask for thermal evaporation of gold. Prior to the gold evaporation, an Ar+ plasma treatment for 20 s followed by the thermal evaporation of titanium with an effective layer thickness of 3 nm was used to enhance the adhesion between the substrate and the gold layer. Then, the gold layer was applied while changing the angle of the surface normal with respect to the evaporation source over time. Therefore, the shadow that was casted by the resist ridges was changing over time, leading to a layer thickness which is a function of the distance from the ridges. The evaporation time was chosen to produce a maximum layer thickness of 100 nm while the starting angle and the change in angle over time were selected to produce a wedge slope of approximately 0.5°. Afterwards, the resist pattern and the gold on top of it was removed using acetone in an ultrasonic bath. Finally, focused gallium ion beam milling was used to create the excitation edge at different wedge heights. A detailed illustration of the fabrication process is in Figure S1 in Supplement 1.

4.2 Experimental setup

We employ a commercial pseudo-heterodyne scattering scanning near-field optical microscope (NeaSNOM, Neaspec GmbH) to measure the SPP field amplitude on a gold wedge sitting on a glass substrate. We used platinum iridium coated tips (ARROW-NCPt, Nanoworld) with a resonance frequency of 285 kHz. The experimental setup is illustrated in Fig. 4. A continuous wave laser beam with a wavelength of 633 nm was split into two arms: the beam in the signal arm is focused onto the SNOM tip oscillating at a frequency Ω by using a parabolic mirror while the reference arm is reflected off a reference mirror oscillating at a frequency M. The beams from the two arms are overlapped and focused onto a photodetector where the interference signal is detected. Since the SNOM tip and the reference mirror oscillate at frequency of Ω and M, respectively, the beam in the signal arm is modulated at integer harmonics of Ω while the beam in the reference arm is at integer harmonics of M. Hence, the detected interference signal is modulated at linear combinations of Ω and M and their integer harmonics, and these modulated signals are further processed with a lock-in amplifier for separating different harmonic signals. For suppressing the background and the direct laser illumination onto a detector, we process preferably the higher harmonics (3^rd or 4^th) of the modulated signal by the tip. The further modulation with the reference mirror splits the main harmonics (Ω) to side bands (4Ω ± M, 4Ω ± 2M etc.), and these side bands allow us not only to suppress further the contribution from the background incident light but also to measure the phase information in addition to the optical absolute amplitude.

5. Conclusion

In conclusion, we have explored analytically and experimentally the short and long range SPP modes in a gold wedge. Among the two, we are most interested in the SR-SPP mode as it enables the plasmon superfocusing effect by which the electromagnetic field is localized both longitudinally, due to the increasing propagation constant with the decreasing wedge height, and transversally, due to the shrinking penetration depth of fields into the substrate and the air with the decreasing wedge height. This electromagnetic field confinement also accompanies with changes in the field amplitude. There is an increase of the amplitude due to the shrinking dimension of the waveguide and the in-phase field oscillation in the metal that further drives the electron oscillation. Contrary, there is also a decrease due to the increasing propagation loss toward the tip. In general, the propagation loss dominates toward the wedge tip due to the increasing imaginary part of the propagation constant. We estimate that the strongest longitudinal confinement effect takes place for a vacuum wavelength of about 526 nm where the mode’s energy density is the heighest in a gold wedge and hence increasing the mode’s effective index faster and more efficient compared with other wavelengths. We have also managed to measure the first 5 half cycle of the SR-SPP mode, and at the 5^th cycle, the half effective wavelength was reduced by 47% compared with the first one. This means the effective index was about 3.5 or the vacuum wavelength is reduced by 3.5 times. This is a clear demonstration of the plasmon superfocusing effect. The gold wedge is a simple structure to analytically model and experimentally explore the SPP modes and their behavior as well as the material model and its validity. Furthermore, there are more aspects that can be studied with such structures such as by using an ultrashort femtosecond pulse, one can study the dispersion of superfocusing effect as well as their dispersion during a propagation and their dispersion compensation. Once achieving a spatially and temporally confined spot with an ultrashort femtosecond pulse at the sharp tip of the wedge, the study can be further expanded to light matter interactions depending on the varying local density of states in the wedge and its enhancement by the superfocusing effect.